3.1 MPT and the CAPM

Why Portfolio Theory Matters on FRM Part I

GARP’s Foundations reading on modern portfolio theory (MPT) and the Capital Asset Pricing Model (CAPM) sits at the core of how risk managers think about priced risk, diversification, and performance attribution. Exam questions routinely ask you to locate a portfolio on the efficient frontier, compute a CAPM-required return, interpret beta, or choose the correct performance ratio for a given risk definition. The arithmetic is straightforward once the definitions are locked in; the traps are mixing total risk with systematic risk and confusing the capital market line with the security market line.

---

Markowitz Mean-Variance Framework

Harry Markowitz’s insight is that investors care about portfolio mean and variance, not the moments of each security in isolation. For a portfolio of weights wi on assets with expected returns μi, covariances σij, and variances σi²:

• Expected return: μp = Σi wi μi
• Variance: σp² = Σi Σj wi wj σij

When returns are imperfectly correlated, portfolio variance falls below the weighted average of stand-alone variances. That reduction is the quantitative content of diversification.

The Efficient Frontier

Plot expected return on the vertical axis and standard deviation on the horizontal axis. The minimum-variance frontier is the set of portfolios that achieve the lowest σp for each target μp. The upper half of that frontier—from the global minimum-variance portfolio upward—is the Markowitz efficient frontier. Rational mean-variance investors hold only efficient portfolios; any interior combination is dominated.

Two-Fund Separation (Risky Assets Only)

With only risky assets and identical beliefs, every efficient portfolio is a combination of two frontier portfolios. Adding a risk-free asset collapses the tangency geometry further: the efficient set becomes the straight line from the risk-free rate through the tangency (market) portfolio—the capital market line discussed below.

Worked Diversification Example

Asset A: μA = 10%, σA = 20%. Asset B: μB = 14%, σB = 30%. Correlation ρAB = 0.20. Equal weights wA = wB = 0.50.

• μp = 0.5(10%) + 0.5(14%) = 12%
• σp² = (0.5)²(0.20)² + (0.5)²(0.30)² + 2(0.5)(0.5)(0.20)(0.30)(0.20)
• σp² = 0.01 + 0.0225 + 0.006 = 0.0385
• σp = √0.0385 ≈ 19.6%

Equal-weighted average volatility would be 25%; realized portfolio risk of about 19.6% shows the diversification benefit from low correlation.

---

CAPM Assumptions and Building Blocks

The CAPM embeds MPT in an equilibrium story. Standard FRM-level assumptions include:

1. Investors are mean-variance optimizers with a common single-period horizon.
2. Homogeneous expectations about means, variances, and covariances.
3. Unlimited risk-free lending and borrowing at the same rate Rf.
4. No taxes, transaction costs, or short-sale constraints (frictionless markets).
5. Assets are infinitely divisible; markets are competitive (price-takers).

Under those assumptions, every investor holds a mix of the risk-free asset and the same market portfolio M of all risky assets. Idiosyncratic risk is diversified away in M, so only systematic risk—sensitivity to M—is compensated.

Beta

βi = Cov(Ri, Rm) / Var(Rm) = (ρi,m σi σm) / σm² = (ρi,m σi) / σm

Beta measures the expected percentage move in asset i for a 1% move in the market. By construction, βM = 1 and βRf = 0. Portfolio beta is the weighted average of constituent betas.

CAPM Expected Return (Security Market Line Algebra)

E(Ri) = Rf + βi[E(Rm) − Rf]

The term E(Rm) − Rf is the market risk premium. Assets plotting above the security market line are underpriced (positive alpha in a CAPM world); assets below are overpriced.

Worked CAPM Example

Rf = 3%, E(Rm) = 11%, so the market premium is 8%. Stock X has σX = 35%, σm = 20%, and ρX,m = 0.60.

• βX = (0.60 × 0.35) / 0.20 = 1.05
• E(RX) = 3% + 1.05 × 8% = 11.4%

If the analyst’s forecast return for X is 13%, implied Jensen-style mispricing relative to CAPM is 13% − 11.4% = +1.6% (undervalued on a CAPM basis).

---

CML versus SML

The CML equation for an efficient combination with weight w in M and (1−w) in Rf is:

E(Rp) = Rf + ([E(Rm)−Rf]/σm) × σp

Individual securities generally do not lie on the CML because their total risk includes diversifiable noise. They are still priced on the SML if only beta is rewarded.

---

Performance Measures Used on the Exam

Sharpe Ratio

Sp = (Rp − Rf) / σp

Use when the portfolio represents the investor’s entire risky wealth (total volatility matters). Higher is better. The market’s Sharpe ratio is the CML slope.

Treynor Ratio

Tp = (Rp − Rf) / βp

Use when the portfolio is one sleeve inside a broader diversified book (only systematic risk remains). Ranking by Treynor can differ from Sharpe when residual risk differs.

Jensen’s Alpha

αp = Rp − [Rf + βp(Rm − Rf)]

Alpha is the vertical distance from the SML. Positive alpha means outperformance versus CAPM-required return over the sample period (before fees, unless returns are net).

Tracking Error and Information Ratio

Tracking error (TE) is the standard deviation of active returns Rp − Rb versus a benchmark b. The information ratio (IR) is:

IR = (Rp − Rb) / TE

(Sometimes written with appraisal alpha in the numerator when a factor model is used.) IR measures active return per unit of active risk—central for evaluating long-only active managers against an index mandate.

Sortino Ratio

Sortino = (Rp − RT) / σd

where RT is a target or minimum acceptable return and σd is downside deviation (volatility of returns below RT only). Sortino ignores upside volatility, which investors do not treat as harmful.

Worked Performance Comparison

Portfolio P: Rp = 15%, σp = 18%, βp = 1.20. Market: Rm = 12%, σm = 15%. Risk-free: Rf = 2%. Benchmark active stats: Rp − Rb = 2.5%, TE = 5%. Downside deviation vs a 3% target: σd = 10%.

P beats the market on Sharpe and shows modest positive Jensen alpha. Whether that is impressive depends on costs, sample length, and whether beta was estimated with noise.

---

Exam Pitfalls

• Using σ on the SML or β on the CML.
• Computing beta as σi/σm without the correlation (only valid if ρ = 1).
• Preferring Sharpe for a sleeve that will be mixed with many other sleeves—Treynor or IR may be more relevant.
• Treating a high Sharpe from concentrated idiosyncratic risk as “better risk management”; MPT says that risk may not be priced and may not survive diversification.

Master the formulas, know which risk denominator each ratio uses, and you will convert most FRM–5 quantitative items into routine arithmetic.